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July  2013, 12(4): 1745-1753. doi: 10.3934/cpaa.2013.12.1745

Remarks on nonlinear equations with measures

Moshe Marcus 1,

1. 

Department of Mathematics, Technion, Haifa 32000, Israel

Received  February 2011 Revised  July 2012 Published  November 2012

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Omega$.
Keywords: Orlicz - Sobolev spaces, absorption term, Semi-linear elliptic equation, weak solution, capacities..
Mathematics Subject Classification: Primary: 35J65, 46E30; Secondary: 46E35, 32U2.
Citation: Moshe Marcus. Remarks on nonlinear equations with measures. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1745-1753. doi: 10.3934/cpaa.2013.12.1745
References:
[1]

D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'' Grundlehren Math. Wissen., 314, Springer, 1996.

[2]

N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz, Ann. Sci. Math. Québec, 12 (1994), 1-23.

[3]

P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires, Ann. Inst. Fourier, 34 (1984), 185-206.

[4]

D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data, Ann. I. H. Poincaré - AN, 22 (2005), 799-815.

[5]

Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation, J. Evolution Eq., 3 (2003), 673-770.

[6]

H. Brezis, Notes unpublished (circa 1970).

[7]

H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.

[8]

Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. An., 8 (1971), 52-75.

[9]

M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'' P. Noordhoff, Groningen, 1961.

[10]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.

[11]

M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal., 144 (1998), 201-231.

[12]

M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption, J. European Math. Soc., 6 (2004), 483-527.

[13]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.

[14]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'' Pitman Research Notes, 353, Longman, 1996.

show all references

References:
[1]

D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'' Grundlehren Math. Wissen., 314, Springer, 1996.

[2]

N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz, Ann. Sci. Math. Québec, 12 (1994), 1-23.

[3]

P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires, Ann. Inst. Fourier, 34 (1984), 185-206.

[4]

D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data, Ann. I. H. Poincaré - AN, 22 (2005), 799-815.

[5]

Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation, J. Evolution Eq., 3 (2003), 673-770.

[6]

H. Brezis, Notes unpublished (circa 1970).

[7]

H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.

[8]

Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. An., 8 (1971), 52-75.

[9]

M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'' P. Noordhoff, Groningen, 1961.

[10]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.

[11]

M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal., 144 (1998), 201-231.

[12]

M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption, J. European Math. Soc., 6 (2004), 483-527.

[13]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.

[14]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'' Pitman Research Notes, 353, Longman, 1996.

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